Lesson 2: Properties of Rigid Transformations

Property

A rigid transformation is a change in the position of a figure that perfectly preserves its size and shape. Two figures are congruent ($\cong$) if and only if one can be mapped exactly onto the other by a sequence of one or more rigid transformations:

1. Translations (slides)
2. Reflections (flips)
3. Rotations (turns)

Examples

• Translation (Slide): A triangle $\Delta ABC$ is moved 5 units to the right and 2 units up to map perfectly onto $\Delta A'B'C'$.
• Reflection (Flip): A triangle $\Delta DEF$ is flipped across the y-axis to create a congruent mirror image, $\Delta D'E'F'$.
• Sequence of Transformations: Pentagon $DEFGH$ is translated 2 units up, then rotated 270° clockwise to produce a congruent pentagon $D''E''F''G''H''$.

Explanation

Rigid transformations (also known as isometries) are simply the "vehicles" we use to drive one shape over to park exactly on top of its clone. Think of it as sliding, flipping, or turning a paper cutout on a desk; the cutout itself remains unchanged. By tracking these movements using prime notation ($A \rightarrow A' \rightarrow A''$), we can prove two shapes are identical without measuring them.

Property

In congruent figures, corresponding parts are the matching angles and sides that occupy the same relative positions.

A congruence statement ($\Delta ABC \cong \Delta DEF$) is valid if and only if the vertex order perfectly reflects the actual correspondence: